The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, ${v·n̅|}_S=0$, S = ∂Ω in weighted $L_p$-Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.

For a bounded domain $\Omega \subset {ℝ}^n$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u+u·\nabla u+\nabla p=f$, $divu=k$, $u_{{|}_{\partial \Omega }}=g$ with $u\in L^q$, $q\ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of...

On présente dans cet exposé des résultats récents de Merle et Raphael sur l’analyse des solutions explosives de l’équation de Schrödinger $L^2$ critique. On s’intéresse en particulier à leur preuve du fait que les solutions d’énergie négative (dont on savait qu’elles explosaient par l’argument du viriel) et dont la norme $L^2$ est proche de celle de l’état fondamental, explosent au régime du “log log”et que ce comportement est stable.

One proves, in the case of piecewise smooth coefficients, that the time derivative of the solution of the so called dam problem is a measure, extending the result proved by the same authors in the case of Lipschitz continuous coefficients.